Section 9.7 Complex Numbers

Objectives Express square roots of negative numbers in terms of i Write complex numbers in the form a + bi Add and subtract complex numbers Multiply complex numbers Divide complex numbers Perform operations involving powers of i

Objective 1: Express Square Roots of Negative Numbers in Terms of i Some equations do not have real-number solutions. For example, x2 = –1 has no real number solutions because the square of a real number is never negative. To provide a solution to this equation, mathematicians have defined the number i so that i2 = –1. The imaginary number i is defined as From the definition, it follows that i2 = –1

Objective 1: Express Square Roots of Negative Numbers in Terms of i Square Root of a Negative Number: For any positive real number b,

EXAMPLE 1 Write each expression in terms of i: Strategy We will write each radicand as the product of –1 and a positive number. Then we will apply the appropriate rules for radicals. Why We want our work to produce a factor of so that we can replace it with i.

EXAMPLE 1 Solution Write each expression in terms of i: After factoring the radicand, we use an extension of the product rule for radicals. d. After factoring the radicand, use an extension of the product and quotient rules for radicals.

Objective 2: Write Complex Numbers in the Form a + bi A complex number is any number that can be written in the form a + bi, where a and b are real numbers and For a complex number written in the standard form a + bi, we call a the real part and b the imaginary part. Some examples of complex numbers written in standard form are Complex numbers of the form a + bi, where b ≠ 0, are also called imaginary numbers.

Objective 2: Write Complex Numbers in the Form a + bi The following illustration shows the relationship between the real numbers, the imaginary numbers, and the complex numbers.

EXAMPLE 2 Write each number in the form a + bi : Strategy We will determine a, the real part, and we will simplify the radical (if necessary) to determine the bi part. Why We can put the two parts together to produce the desired a + bi form.

EXAMPLE 2 Write each number in the form a + bi : Solution

Objective 3: Add and Subtract Complex Numbers Adding and subtracting complex numbers is similar to adding and subtracting polynomials. To add complex numbers, add their real parts and add their imaginary parts. To subtract complex numbers, add the opposite of the complex number being subtracted.

EXAMPLE 3 Perform each operation. Write the answers in the form a + bi : Strategy To add the complex numbers, we will add their real parts and add their imaginary parts. To subtract the complex numbers, we will add the opposite of the complex number to be subtracted. Why We perform the indicated operations as if the complex numbers were polynomials with i as a variable.

EXAMPLE 3 Perform each operation. Write the answers in the form a + bi : Solution

EXAMPLE 3 Perform each operation. Write the answers in the form a + bi : Solution

Objective 4: Multiply Complex Numbers Since imaginary numbers are not real numbers, some properties of real numbers do not apply to imaginary numbers. For example, we cannot use the product rule for radicals to multiply two imaginary numbers. Multiplying complex numbers is similar to multiplying polynomials.

EXAMPLE 4 Multiply: Strategy To multiply the imaginary numbers, we will first write in form. Then we will use the product rule for radicals. Why We cannot immediately use the product rule for radicals because it does not apply when both radicands are negative.

EXAMPLE 4 Multiply: Solution

Objective 5: Divide Complex Numbers Before we can discuss division of complex numbers, we must introduce an important fact about complex conjugates. The complex numbers are called complex conjugates. In general, the product of the complex number a + bi and its complex conjugate a − bi is the real number a2 + b2.

Objective 5: Divide Complex Numbers Division of Complex Numbers: To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator.

EXAMPLE 8 Divide. Write the answers in the form a + bi. Strategy We will build each fraction by multiplying it by a form of 1 that uses the conjugate of the denominator. Why This step produces a real number in the denominator so that the result can then be written in the form a + bi.

EXAMPLE 8 Solution Divide. Write the answers in the form a + bi. a. We want to build a fraction equivalent to that does not have i in the denominator. To make the denominator, 6 + i, a real number, we need to multiply it by its complex conjugate, 6 − i. It follows that should be the form of 1 that is used to build .

EXAMPLE 8 Solution Divide. Write the answers in the form a + bi. b. We can make the denominator of a real number by multiplying it by the complex conjugate of , which is . It follows that should be the form of 1 that is used to build .

EXAMPLE 8 Solution Divide. Write the answers in the form a + bi. Combine like terms in the numerator and denominator.

Objective 6: Perform Operations Involving Powers of i Powers of i: If n is a natural number that has a remainder of R when divided by 4, then in = iR. The powers of i produce an interesting pattern:

EXAMPLE 11 Strategy We will examine the remainder when we divide the exponents 55 and 98 by 4. Why The remainder determines the power to which i is raised in the simplified form.

EXAMPLE 11 Solution a. We divide 55 by 4 and get a remainder of 3. Therefore, i55 = i3 = –i b. We divide 98 by 4 and get a remainder of 2. Therefore, i98 = i2 = –1